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On Quantum Tunnelling with and Without Decoherence and the Direction of Time

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On Quantum Tunnelling with and Without Decoherence and the Direction of Time On the effect of decoherence on quantum tunnelling. By A.Y. Klimenko † The University of Queensland, SoMME, QLD 4072, Australia (SN Applied Sciences, 2021, 3:710) Abstract This work proposes a series of quantum experiments that can, at least in principle, allow for examining microscopic mechanisms associated with decoherence. These experiments can be interpreted as a quantum-mechanical version of non-equilibrium mixing between two volumes separated by a thin interface. One of the principal goals of such experiments is in identifying non-equilibrium conditions when time-symmetric laws give way to time-directional, irreversible processes, which are represented by decoher- ence at the quantum level. The rate of decoherence is suggested to be examined indirectly, with minimal intrusions | this can be achieved by measuring tunnelling rates that, in turn, are affected by decoherence. Decoherence is understood here as a general process that does not involve any significant exchanges of energy and governed by a particular class of the Kraus operators. The present work analyses different regimes of tunnelling in the presence of decoherence and obtains formulae that link the corresponding rates of tunnelling and decoherence under different conditions. It is shown that the effects on tunnelling of intrinsic decoherence and of decoherence due to unitary interactions with the environment are similar but not the same and can be distinguished in experiments. Keywords: decoherence, quantum tunnelling, non-equilibrium dynamics 1. Introduction The goal of this work is to consider experiments that can, at least in principle, examine time-directional quantum effects in an effectively isolated system. Such experiments need to be conducted somewhere at the notional boundary between the microscopic quantum and macroscopic thermodynamic worlds, that is we need to deal with quantum systems that can exhibit some degree of thermodynamic behaviour. At the quantum level, this corresponds to persisting decoherence, which is, perhaps, the most fundamental irreversible process that we are aware of | it takes place at the smallest scales, increases entropy [1] and, expectedly, induces various macroscopic effects associated with the thermodynamic arrow of time [2]. A large volume of literature is dedicated to decoherence, which may involve both intrinsic [3{6] and environmental [6{13] mechanisms. The present work examines a problem that, at least conceptually, can become an experiment probing the direction of time. This problem represents a quantum-mechanical version of non-equilibrium mixing between two volumes separated by a thin interface. In this quantum version of the classical problem, particles tunnel through the interface and, at the same time, are subject to the omnipresent influence of quantum decoherence, which, presumably, is the fundamental mechanism enacting non-equilibrium, time-directional effects in the macroscopic world [14]. arXiv:2011.12955v2 [quant-ph] 19 Sep 2021 In quantum experiments, one has to face another fundamental difficulty | interferences from the environment and measurements. Environmental interferences can overwhelm intrinsic mechanisms of de- coherence, while quantum measurements routinely cause decoherences and collapses (which are interpreted here as defined in the Appendix of Ref. [15]) instead of observing these decoherences and collapses without interfering. It appears, however, that, under the conditions examined in this work, decoherence affects the rate of quantum tunnelling and, therefore, can be characterised by the tunnelling rates without measuring decoherence directly. Among many formulations of tunnelling problems [16{22], we select one that has a transparent and, at the same time, sufficiently general solution. For this formulation involving quantum † email: [email protected] SN Appl. Sci., 2021, 3:710 TEX Paper 2 A.Y. Klimenko tunnelling through a high potential barrier under non-equilibrium conditions, we examine mechanisms that may be ultimately responsible for the direction of time. Conducting such experiments is not easy but seems possible even under the current level of technology. Conceptually and technically similar ex- periments have been performed in the past [18, 20, 22{24]. These experiments investigated mesoscopic decoherence in context of the Aharonov-Bohm effect [23], proton tunnelling under thermal bath condi- tions [18], the effect of invasive frequent measurements on quantum tunnelling [20] (i.e. the quantum Zeno effect [25]). Our main interest is in examining decoherence by using tunnelling but, unlike in the previous of experiments, under conditions that avoid direct interferences from the environment and measurements, and screen the experiment from a supposed direct influence of the temporal boundary conditions imposed on the universe. † This manuscript is organised as follows. Section 2 briefly reviews the interpretation of the arrow of time from a philosophical perspective pointing to the ideas of Hans Reichenbach as a source of critical thinking about the time that is relevant to the present work. The readers, who are interested in quantum mechanics and non-equilibrium dynamics more than in philosophical issues, can omit this section at first reading. Section 3 introduces the tunnelling problem and, in the context of this problem, discusses emergence of the arrow of time. Section 4 reviews different time-asymmetric and time-symmetric interpretations of quantum mechanics, in particular the two-state vector formalism [26{28]. Section 5 examines tunnelling in absence of decoherence, while Section 6 investigates the influence of decoherence on the tunnelling rates. Section 7 discusses conduct of experiments based on the results of this work. Section 8 summarises our conclusions. More extensive derivations of asymptotic tunnelling rates are presented in Appendix A and a brief consideration of the problem from the perspective of the theory of environmental decoherence [7] is given in Appendix B. 2. Discrimination of the past and the future from a philosophical perspective It is well known that the most important physical laws | those of classical, relativistic and quantum mechanics | are time-symmetric, but our experience of physical reality strongly discriminates the past and the future. The observed arrow of time is reflected in the second law of thermodynamics, which permits entropy increases but bans reduction of entropy in isolated thermodynamic systems. While the Boltzmann time hypothesis, which suggests that the perceived arrow of time and the direction of entropy increase must be the same (i.e. connected at some fundamental level), may be striking at first, but after some thinking over the issue, most people tend to arrive at the same conclusion. Since Ludwig Boltzmann [15, 29], the overall conditions prevailing in the universe (or its observable part) have been thought to be responsible for this temporal asymmetry. In modern physics, the increasing trend for entropy is commonly explained by the asymmetry of temporal boundary conditions imposed on the universe, i.e. by low-entropy conditions at the time of Big Bang [30]. This explanation is called the past hypothesis by Albert [31] and in other publications. There are no doubts that the past conditions existing in the universe are very important. The pertinent question, however, is not whether these conditions are important, but whether the direct influence of the initial conditions imposed on the universe is sufficient to explain all time-directional phenomena observed in experiments. A number of publications seem to be content with the sufficiency of the special initial conditions in the early universe to explain all entropy increases in thermodynamically isolated systems, even if it is presumed that all laws of physics are time-symmetric [31, 32]. The alternative view is that the past hypothesis is important but, on its own, is insufficient to fully explain entropy increases required by the second law. This view can be traced to the principle of parallelism of entropy increase, which was introduced by Hans Reichenbach [33], and further explained, evaluated and extended by Davies [34], Sklar [35] and Winsberg [36, 37]. The Reichenbach principle concurs that initial conditions imposed on the universe can explain many effects associated with entropy increase; nor does it deny that entropy can fluctuate. The initial conditions imposed on the universe may explain why entropy tends to increase more often than to decrease in semi-isolated thermodynamic subsystems (branches) but, † The key question investigated in the present work | whether the intrinsic and environmental mechanisms of decoherence can be distinguished in experiments | has been repeatedly raised in the past: for example, by G.N. Hatsopoulos and G.P. Beretta in "Where is the entropy challenge?" [AIP Conf. Proc. 1033, 34-54 (2008)]. SN Appl. Sci., 2021, 3:710 On the effect of decoherence on quantum tunnelling. 3 assuming that all governing physical laws are time-symmetric, these initial conditions do not explain the persistence and consistency of this increase (this, of course, does not exclude the existence of fluctuations of entropy but indicates that, according to the fluctuation theorem, entropy increases over a given time interval are consistently more likely than entropy decreases [38]). Consider a system that is isolated from the rest of the universe without reaching internal equilibrium: would such a system demonstrate conventional thermodynamic behaviour, or would its entropy increase terminate
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